%5E2.jpeg "(5-x^2)^2")
Expanding the Expression (5-x^2)^2
The expression (5-x^2)^2 represents the square of a binomial. To expand this expression, we can use the following techniques:
1. Using the FOIL method
FOIL stands for First, Outer, Inner, Last. It's a mnemonic device to help remember the steps involved in multiplying two binomials.
- First: Multiply the first terms of each binomial: 5 * 5 = 25.
- Outer: Multiply the outer terms of the binomials: 5 * (-x^2) = -5x^2.
- Inner: Multiply the inner terms of the binomials: (-x^2) * 5 = -5x^2.
- Last: Multiply the last terms of each binomial: (-x^2) * (-x^2) = x^4.
Now, add all the terms together: 25 - 5x^2 - 5x^2 + x^4 = x^4 - 10x^2 + 25
2. Using the Square of a Binomial Formula
The formula for the square of a binomial is: (a - b)^2 = a^2 - 2ab + b^2
In our case, a = 5 and b = x^2. Substituting these values into the formula, we get:
(5 - x^2)^2 = 5^2 - 2(5)(x^2) + (x^2)^2
Simplifying this expression leads to:
x^4 - 10x^2 + 25
Conclusion
Both methods provide the same result: (5 - x^2)^2 = x^4 - 10x^2 + 25. Understanding these methods allows you to efficiently expand any binomial squared.